50 
ON THE TRANSFORMATION OF THE EQUATION OF A 
[566 
566] 
We find 
-Pai + Qßi + -Ryi = («>•• •) («1, ßi> Yi) 2 > =0, 
Pi 
Pa 2 + Q/3 2 -F Ry 2 = (a, ...) (a 1} ft, yj) (a 2 , ft, 7 2 ) (a ia2 + ftft + YiYs)» = °> 
Pi 
and thence 
P Q ■ R = /SjYs /ftyi • Yi a 2 Ya^i • a. 2 ft 
= a : /3 : y , 
or say 
p, g, P = M, № 
we have thus the equations 
Pi 
and joining hereto 
h 
■ 3 ) 
( a i, ßi, Yi) ~ 
b- 
h / ) 
II 
£ 
« 
f 
■ -s 
(«1, A, Vi) = 8fi, 
(A, 
B, C)(a 1} ft, Yi) = 0, 
ain the equation 
1 
Pi’ 
h , 
9 > A 
= 0, 
h 
b ~b 
Pi 
f > b 
9 
f i c 
G 
Pi 
A 
B , 
G , 0 
1 
Pi 
= (a,...)(a 2 
ft) Y2) 2 » 
and in like manner writing 
we have the same equation for p 2 \ wherefore p 1} p 2 are the roots of the quadric equation 
1 
a--, h , g , A 
r 
h , b-~ f , B 
r 
g > f , c — ^, c 
'A , B , G , 
= 0. 
Moreo\ 
determinani 
say, a 1} ft, 
question; a 
But we 
that is, 
and in the 
Hence 
where it w: 
terms (X, ] 
in 8x, 8y, 8 
coefficients 
radii of cur 
values of tl 
terms in X 
Z is of the 
in Z 2 of the 
0